Abstract
The Noether theorem and its inverse theorem for the nonlinear dynamical systems with nonstandard Lagrangians are studied. In this paper, two kinds of nonstandard Lagrangians, namely exponential Lagrangians and power-law Lagrangians, are discussed. For each case, the Hamilton principle based on the action with nonstandard Lagrangians is established, the differential equations of motion for the dynamical systems with nonstandard Lagrangians are obtained, and two basic formulae for the variation in Hamilton action with nonstandard Lagrangians are derived. The definitions and the criteria of the Noether symmetric transformations and the Noether quasi-symmetric transformations are given. The Noether theorem and its inverse theorem are established, which reveal the intrinsic relation between the symmetry and the conserved quantity for the systems with nonstandard Lagrangians. Two examples are given to illustrate the application of the results.
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